An analog of the Schwarz lemma for locally quasiconformal automorphisms of the unit disk

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• 作者：S. Yu. Graf
• 关键词：locally quasiconformal mapping ; growth theorem ; Schwarz’s lemma
• 刊名：Russian Mathematics (Iz VUZ)
• 出版年：2014
• 出版时间：November 2014
• 年：2014
• 卷：58
• 期：11
• 页码：74-79
• 全文大小：508 KB
• 参考文献：1. Goluzin, G. M. / Geometric Theory of Functions of one Complex Variable (Nauka, Moscow, 1966) [in Russian].
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  4. Anderson, G. D., Vamanamurthy, M. K., Vuorinen, M. / Conformal Invariants, Inequalities and Quasiconformal Maps (Wiley & Sons, N. Y., 1997).
  5. Vasil’ev, A. / Moduli of Families of Curves for Conformal and Quasiconformal Mappings (Springer, Berlin-N. Y., 2002). CrossRef
  6. Ahlfors, L. V. / Lectures on Quasiconformal Mappings (D. Van Nostrand Company, Toronto-New York-London, 1966; Mir,Moscow, 1969).
  7. Graf, S. Yu., Eyelangoli, O. R. “On Distortion of Modules of Doubly-Connected Domains Under Locally Quasiconformal Mappings,-in / Application of Functional Analysis to Approximation Theory (Tver, 2009), pp. 34-3 [in Russian].
  8. Belinskii, P. P. / General Properties of Quasiconformal Maps (Nauka, Novosibirsk, 1974) [in Russian].
  9. Sheil-Small, T. “Constants for Planar Harmonic Mappings,-J. London Math. Soc. 42, No. 2, 237-48 (1990). CrossRef
  10. Graf, S. Yu. “Growth Theorems in Classes of Normalized Locally Quasiconformal Mapppings,-Probl. Anal. 2(20), No. 1, 3-0 (2013) [in Russian].
  11. Clunie, J., Sheil-Small, T. “Harmonic Univalent Functions,-Ann. Acad. Sci. Fenn., Ser. A. I.Math. 9, 3-5 (1984). CrossRef
  
• 作者单位：S. Yu. Graf (1)
  
  1. Tver State University, ul. Zhelyabova 33, Tver, 170000, Russia
  
• ISSN：1934-810X

文摘

We obtain sharp estimates for the module of functions in the classes of normalized locally quasiconformal authomorphisms of the unit disk with given majorants of M. A. Lavrent’ev’s characteristic. The estimates are analogs of Schwarz’s lemma and A. Mori’s theoremand they imply the classical growth theorems for quasiconformal authomorphisms of the disk. In the classes we also prove sharp estimates of the conformal radius and the radius of covering disk. The main results are obtained by methods of extremal lengths and symmetrization.